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H-cobordism

A bordism, where is a compact manifold whose boundary is the disjoint union of closed manifolds which are deformation retracts (cf. Deformation retract) of . The simplest example is the trivial -cobordism

Two manifolds and are said to be -cobordant if there is an -cobordism joining them.

If is an -cobordism such that , , are simply-connected differentiable (or piecewise-linear) manifolds and , then is diffeomorphic (or piecewise-linearly isomorphic) to : and therefore (the -cobordism theorem [4]). Thus, proving the isomorphism reduces to providing an -cobordism, which can be achieved by methods of algebraic topology. For this reason, this theorem is basic in passing from the homotopy classification of simply-connected manifolds to their classification up to a diffeomorphism (or a piecewise-linear isomorphism). Thus, if , , is a compact differentiable manifold with simply-connected boundary, then it is diffeomorphic to the disc . If , , is a manifold that is homotopy equivalent to the sphere , then it is homeomorphic (and even piecewise-linearly isomorphic) to (the generalized Poincaré conjecture).

The -cobordism theorem allows one to classify the differentiable structures on the sphere , [6], and also on the homotopy type of an arbitrary closed simply-connected manifold , [1].

In the case of an -cobordism with there is, in general, no diffeomorphism from to .

All -cobordisms with and fixed are classified by a certain Abelian group, namely the Whitehead group of the group . Corresponding to a given -cobordism is an element of that is an invariant of the pair ; it is denoted by and is called the torsion (sometimes the Whitehead torsion) of the given -cobordism. If (or, equivalently, ), then the -cobordism is called an -cobordism. If is an -cobordism such that , then vanishes if and only if (the -cobordism theorem). The -cobordism theorem is a special case of this theorem in view of the fact that . The -cobordism theorem is also true for topological manifolds [9].

For an -cobordism , the torsion is defined along with ; if the given -cobordism is orientable, then , where and the element is conjugate to in the group . In particular, if is finite and Abelian, .

If two -cobordisms and are glued along to the -cobordism , then

If two copies of are glued along , where is odd and , then one obtains an -cobordism , where when there is no diffeomorphism from to , that is, when does not imply that the -cobordism connecting them is trivial.

If is a closed connected manifold and , then there exists for any an -cobordism with . If and (with ) have the same torsion , then relative to . When is even and is finite, there is a finite set of distinct manifolds that are -cobordant with . This is not the case when is odd.

If two homotopy-equivalent manifolds and are imbedded in , with sufficiently large, and their normal bundles are trivial, then and are -cobordant. If, moreover, and are of the same simple homotopy type, that is, if the torsion of this homotopy equivalence vanishes, then .

If is an -cobordism and is a closed manifold, then there is an -cobordism with , where is the Euler characteristic of . If and , then

In particular, ; furthermore, two closed manifolds and of the same dimension are -cobordant if and only if .

The -cobordism structure has not been completely elucidated for (1978). Thus there is the following negative result [8]: There exists an -cobordism , where is the four-dimensional torus, for which there is no diffeomorphism from to ; since , this means that the -cobordism theorem fails for .